Anisotropic improved Leray–Trudinger inequality

We establish a Leray–Trudinger type inequality in the anisotropic setting induced by a strongly convex Finsler norm F . The result generalizes classical exponential integrability inequalities for Sobolev functions to the framework of anisotropic Sobolev spaces W01,n(Ω), where the standard Euclidean norm is replaced by F and the associated polar norm F ∘. In the general anisotropic setting we obtain an admissible exponential constant depending on the anisotropic Hardy difference. Moreover, in the class of anisotropically radial functions, we identify the sharp exponential constant, in the sense that the corresponding estimate holds up to the critical threshold and fails for every larger constant, in the spirit of Moser’s sharp inequality.